Chapter III.—Demonstration Defined.

Similarly, also, all men will admit
that demonstration is discourse,37053705 It is necessary to read λόγον
here, though not in the text, on account of ἐκπορίζοντα
which follows; and as εὔλογον
εἷναι λόγον
occurs afterwards, it seems better to retain δὔλογον
than to substitute λόγον for
it. agreeable to reason, producing belief in points disputed,
from points admitted.

Now, not only demonstration and belief and knowledge,
but foreknowledge also, are used in a twofold manner. There is that
which is scientific and certain, and that which is merely based on
hope.

In strict propriety, then, that is called demonstration
which produces in the souls of learners scientific belief. The other
kind is that which merely leads to opinion. As also, both he that is
really a man, possessing common judgment, and he that is savage and
brutal,—each is a man. Thus also the Comic poet said that
“man is graceful, so long as he is man.” The same holds
with ox, horse, and dog, according to the goodness or badness of the
animal. For by looking to the perfection of the genus, we come to those
meanings that are strictly proper. For instance, we conceive of a
physician who is deficient in no element of the power of healing, and a
Gnostic who is defective in no element of scientific knowledge.

Now demonstration differs from syllogism; inasmuch
as the point demonstrated is indicative of one thing, being one and
identical; as we say that to be with child is the proof of being no longer
a virgin. But what is apprehended by syllogism, though one thing, follows
from several; as, for example, not one but several proofs are adduced
of Pytho having betrayed the Byzantines, if such was the fact. And to
draw a conclusion from what is admitted is to syllogize; while
to draw a conclusion from what is true is to demonstrate.

So that there is a compound advantage of demonstration:
from its assuming, for the proof of points in question, true premisses,
and from its drawing the conclusion that follows from them. If the
first have no existence, but the second follow from the first, one has
not demonstrated, but syllogized. For, to draw the proper conclusion
from the premisses, is merely to syllogize. But to have also each of
the premisses true, is not merely to have syllogized, but also to have
demonstrated.

And to conclude, as is evident from the word, is to
bring to the conclusion. And in every train of reasoning, the point
sought to be determined is the end, which is also called the
conclusion. But no simple and primary statement is termed a syllogism,
although true; but it is compounded of three such, at the
least,—of two as premisses, and one as conclusion.

Now, either all things require demonstration, or some of
them are self-evident. But if the first, by demanding the demonstration
of each demonstration we shall go on ad infinitum; and so
demonstration is subverted. But if the second, those things which are
self-evident will become the starting points [and fundamental grounds]
of demonstration.

In point of fact, the philosophers admit that the first
principles of all things are indemonstrable. So that if there is
demonstration at all, there is an absolute necessity that there be
something that is self-evident, which is called primary and
indemonstrable.

Consequently all demonstration is traced up to
indemonstrable faith.37063706 [We begin, that is, with axioms: and he
ingeniously identifies faith with axiomatic truth. Hence the faith not
esoteric.]

It will also turn out that there are other starting
points for demonstrations, after the source which takes its rise in
faith,—the things which appear clearly to sensation and
understanding. For the phenomena of sensation are simple, and incapable
of being decompounded; but those of understanding are simple, rational,
and primary. But those produced from them are compound, but no less
clear and reliable, and having more to do with the reasoning faculty
than the first. For therefore the peculiar native power of reason,
which we all have by nature, deals with agreement and disagreement. If,
then, any argument be found to be of such a kind, as from points
already believed to be capable of producing
560belief in what is not yet believed, we shall aver that
this is the very essence of demonstration.

Now it is affirmed that the nature of demonstration, as
that of belief, is twofold: that which produces in the souls of the
hearers persuasion merely, and that which produces knowledge.

If, then, one begins with the things which are evident
to sensation and understanding, and then draw the proper conclusion, he
truly demonstrates. But if [he begin] with things which are only
probable and not primary, that is evident neither to sense nor
understanding, and if he draw the right conclusion, he will syllogize
indeed, but not produce a scientific demonstration; but if [he draw]
not the right conclusion, he will not syllogize at all.

Now demonstration differs from analysis. For each one of
the points demonstrated, is demonstrated by means of points that are
demonstrated; those having been previously demonstrated by others; till
we get back to those which are self-evident, or to those evident to
sense and to understanding; which is called Analysis. But demonstration
is, when the point in question reaches us through all the intermediate
steps. The man, then, who practices demonstration, ought to give great
attention to the truth, while he disregards the terms of the premisses,
whether you call them axioms, or premisses, or assumptions. Similarly,
also, special attention must be paid to what suppositions a conclusion
is based on; while he may be quite careless as to whether one choose to
term it a conclusive or syllogistic proposition.

For I assert that these two things must be attended to
by the man who would demonstrate—to assume true premisses, and to
draw from them the legitimate conclusion, which some also call
“the inference,” as being what is inferred from the
premisses.

Now in each proposition respecting a question there must
be different premisses, related, however, to the proposition laid down;
and what is advanced must be reduced to definition. And this definition
must be admitted by all. But when premisses irrelevant to the
proposition to be established are assumed, it is impossible to arrive
at any right result; the entire proposition—which is also called
the question of its nature—being ignored.

In all questions, then, there is something which is
previously known,—that which being self-evident is believed
without demonstration; which must be made the starting point in their
investigation, and the criterion of apparent results.

3705 It is necessary to read λόγον
here, though not in the text, on account of ἐκπορίζοντα
which follows; and as εὔλογον
εἷναι λόγον
occurs afterwards, it seems better to retain δὔλογον
than to substitute λόγον for
it.

3706 [We begin, that is, with axioms: and he
ingeniously identifies faith with axiomatic truth. Hence the faith not
esoteric.]